All Questions
Tagged with classical-mechanics noethers-theorem
117
questions
57
votes
6
answers
19k
views
What symmetry causes the Runge-Lenz vector to be conserved?
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...
- 5,725
33
votes
3
answers
6k
views
Why is Noether's theorem important?
I am just starting to wrap my head around analytical mechanics, so this question might sound weird or trivial to some of you.
In class I have been introduced to Noether's theorem, which states that ...
- 490
30
votes
6
answers
8k
views
Noether Theorem and Energy conservation in classical mechanics
I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
- 10.1k
28
votes
2
answers
9k
views
Invariance of Lagrangian in Noether's theorem
Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$.
However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
- 2,890
27
votes
2
answers
6k
views
Why does the classical Noether charge become the quantum symmetry generator?
It is often said that the classical charge $Q$ becomes the quantum generator $X$ after quantization. Indeed this is certainly the case for simple examples of energy and momentum. But why should this ...
- 7,936
26
votes
1
answer
18k
views
Constants of motion vs. integrals of motion vs. first integrals
Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
user17116
20
votes
2
answers
2k
views
Layman's version of Noether's Theorem (or the intuition behind it)
As part of a science project, I have to give a presentation to my classmates about a topic of my choice (within some constraints) - and I chose symmetry, and it's importance in physics. One important ...
- 499
17
votes
6
answers
3k
views
What symmetry is responsible for the amplitude independence of the period of a simple harmonic oscillator?
In the ICTP lectures of Y. Grossman: Standard Model 1, in about minute 54:00, he leaves an informal homework for the students. He ask to find the symmetry related to the conservation of the amplitude ...
- 5,568
15
votes
5
answers
3k
views
Why does time-translational symmetry imply that energy (and not something else) is conserved?
I'm trying to understand Noether's theorem from an intuitive perspective. I know that time-translational symmetry implies the conservation of energy. Is it possible to convince oneself that time-...
- 253
13
votes
6
answers
1k
views
(A modification to) Jon Pérez Laraudogoita’s "Beautiful Supertask" — What assumptions of Noether's theorem fail?
I am curious about the following (physically unrealizable) scenario involving a supertask described here: https://plato.stanford.edu/entries/spacetime-supertasks/#ClasMechSupe. The original paper is ...
- 8,640
9
votes
2
answers
1k
views
Why are symmetries in phase space generated by functions that leave the Hamiltonian invariant?
Hamilton's equation reads
$$ \frac{d}{dt} F = \{ F,H\} \, .$$
In words this means that $H$ acts on $T$ via the natural phase space product (the Poisson bracket) and the result is the correct time ...
- 10.1k
9
votes
3
answers
3k
views
Noether's theorem and time-dependent Lagrangians
Noether's theorem says that if the following transformation is a symmetry of the Lagrangian
$$t \to t + \epsilon T$$
$$q \to q + \epsilon Q.$$
Then the following quantity is conserved
$$\left( \...
- 6,425
9
votes
2
answers
604
views
How general are Noether's theorem in classical mechanics?
I'm going through the derivations of Noether's theorems and I have several criticisms as to how they are presented in popular sources (note that I'm only referring to classical mechanics here and not ...
- 1,619
8
votes
4
answers
2k
views
Noether's theorem for space translational symmetry
Imagine a ramp potential of the form $U(x) = a*x + b$ in 1D space. This corresponds to a constant force field over $x$. If I do a classical mechanics experiment with a particle, the particle behaves ...
- 1,222
8
votes
2
answers
7k
views
Explicit time dependence of the Lagrangian and Energy Conservation
Why is energy (or in more general terms,the Hamiltonian) not conserved when the Lagrangian has an explicit time dependence?
I know that we can derive the identity:
$\frac{d \mathcal{H}}{d t} = - {\...
- 1,487